Jacobi transformation conjecture for VOA characters

Let VV be a rational vertex operator algebra of CFT type, let MM be an irreducible VV-module, and let h{\mathfrak h} and LL^{\circ} be the spaces defined in the source, with LL^{\circ} the dual lattice of the lattice LL. For vhv\in{\mathfrak h} and αL\alpha\in L^{\circ}, write χM(v,τ)\chi_M(v,\tau) for the character associated with MM. Jacobi transformation conjecture.

χM(v+ατ,τ)=e2πi(v,α)πi(α,α)τχM(v,τ).\chi_M(v+\alpha\tau,\tau)=e^{-2\pi i(v,\alpha)-\pi i(\alpha,\alpha)\tau}\chi_M(v,\tau).

This predicts the elliptic transformation law for the characters of irreducible modules and is part of the expected Jacobi-form behavior of VOA characters. The source gives no further information about its resolution.

Sources & referencesView supporting material

Primary source

Chongying Dong, Kefeng Liu and Xiaonan Ma, “Elliptic genus and vertex operator algebras”, arXiv:math/0201135 (2002).

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